Questions tagged [cardinals]

This tag is for questions about cardinals and related topics such as cardinal arithmetics, regular cardinals and cofinality. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

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Cardinality of a nested set

What is the cardinality of {a,{b,{c,d,e},{f,{g}}}}? I think it's 2 because the set length of the broadest {} is 2, but I am not sure if the set being nested will affect the cardinality.
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Recently while trying to prove the Schröder–Bernstein theorem, I stumbled upon a fact that, if $m$ and $n$ are two cardinal numbers such that atleast one of them is infinite, then $m+n= \max\{m,n\}$. ...
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Let $X,Y$ be the collections of all finite subsets of $A$ and $B$ respectively. Assume $|X| \le |Y|$. Is it true that $|A| \le |B|$?

In proving the dimension of infinite-dimensional vector space is well-defined, I come across below question. Let $A,B$ be sets. Let $X,Y$ be the collections of all finite subsets of $A$ and $B$ ...
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Selection function for all non-empty subsets of $\mathrm{R}$ [duplicate]

Task is to find a function, which for any non-empty subset of the reals, gives an element of this subset. For this easy looking problem, something none of the ideas worked, what my IT-accustomed mind ...
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the set of nonrecursive sequences of pos. int. is uncountable

A sequence $a_{n}$ of positive integers is called recursive if there is a positive integer k such that $a_{n} = x_{1}a_{n−1} + ... + x_{k}a_{n−k}$ holds for all $n > k$, where $x_{1}, ..., x_{k}$ ...
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Numbers/cardinal numbers [closed]

so i had two classes this semester abstract algebra and philosophy of math. I enjoyed them both but there is one thing that i have been trying to understand all the semester and just can't get ...
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Necessary and sufficient condition on the cardinal number of quotient set

I'm studying for an exam, and I have found question that I'm not sure about how to solve it. The question is: Let A be finite set. Let R be equivalence relation on A. 1.Write necessary and sufficient ...
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Several forcing axioms imply $2^{\aleph_0 }= \aleph_2$. What about $2^{\aleph_1}$?

On the one hand, it seems intuitive that $2^{\aleph_1 }> 2^{\aleph_0}$, because $\aleph_1 > \aleph_0$. However, I also know that, like many things involving the continnum function, that's ...
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What is the cardinality of the set of untyped $\lambda$-calculus functions? I know it’s an infinity, but is it countable or uncountable?

What is the cardinality of the set of untyped $\lambda$-calculus functions? I know it’s an infinity, but is it countable or uncountable? Let $\Lambda$ be the set of untyped $\lambda$-calculus ...
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Concerning $\kappa+\kappa=\kappa$ for any infinite cardinal $\kappa$

In Introduction to Set Theory by Hrbacek and Jech, on page 94, the authors state that $\aleph_0+\aleph_0=\aleph_0$, and the Axiom of Choice implies that $\kappa+\kappa=\kappa$ for any infinite ...
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When does proof by contradiction occur in Cantor's diagonalization proof?

Cantor's diagonalization argument says that given a list of the reals, one can choose a unique digit position from each of those reals, and can construct a new real that was not previously listed by ...
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What is the cardinality of countable union of infinite sets?

Let $X=\underset{n\in\mathbb{N}}{\bigcup} X_n$ and each $X_n$ has infinite cardinality $\alpha$. What is the cardinality of $X$ ? Is this $\aleph_0\alpha$ ?
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Bijection between $\{0,1\}^\omega$ and the set of positive integers

I have studied the first sections of "Munkres, J.R., Topology". In section 7 I find a theorem (7.7) that states that the set $\{0,1\}^\omega$ ($=\{0,1\}\times\{0,1\}\times\cdots$), i.e., the ...
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Proof that the cardinality of an uncountably infinite set $S$ is equal to the cardinality of $S\setminus \{x\}$, $x$ being an element of $S$.

I have to prove that for some uncountably infinite set $S$ and $x \in S$ the cardinality of $S$ is equal to the cardinality of $S \setminus \{x\}$. I have proven that this is the case for countably ...
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Do hypercontinuous fields exist?

"Hypercontinuity" is a cardinality of a continuous set's power set (set of all subsets). When talking about fields, I mean the cardinality of field's set. At first glance there is nothing ...
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How to prove and define amount of countably infinite set in the naturals numbers

I have the following question Picture of the question What is the cardinality of $D=\{A \subseteq \Bbb N \mid \vert A \vert = \aleph_0 \land \vert \Bbb N \setminus A \vert = \aleph_0 \}$? I do ...
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I am trying to find a source that states, it is consistent with ZFC that weakly inaccessible cardinal does not exist. Can I please get some sources? It was quite hard for me to find such. Indeed, I ...
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$(1*...*1)(n) =\# \{(m_1,...,m_k)\in (\mathbb{N}\setminus \{0\})^k : m_1...m_k = n\}$
We call an arithmetic function any element of $\mathcal{F} (\mathbb{N}\setminus \{0\} , \mathbb{C})$. We endow $\mathcal{F} (\mathbb{N} \setminus \{0\}, \mathbb{C})$ binary operation convolution and ...
How to construct a one-to-one function from $\mathbb{R}$ to a perfect set $P$?
Let, $P$ be a non empty perfect subset of $\mathbb{R}$ . Then, as an application of Baire Catagory theorem, I can show that the set $P$ is uncountable. As we know any uncountable subset of \$\mathbb{R}...